Question

In the following exercise, find the inverse of each function.

$f\left(x\right)=x^3+6$

Short answer

The inverse of the given function is$f^{-1}\left(x\right)=\sqrt[3]{x-6}$

Step-by-step solution

Step 1. Given information  

The given function is$f\left(x\right)=x^3+6$

Step 2.  Interchange the variable x and y

Substituting $f\left(x\right)=y$, we get:

$y=x^3+6$

interchanging the variable $x$and $y$, we get:

$x=y^3+6$

Step 3. Solve for y

Solving for $y$, we get:

$$\begin{gathered} x=y^3+6 \\ {y}^3=x-6 \\ y=\sqrt[3]{\left(x-6\right)} \end{gathered}$$

substituting $y=f^{-1}\left(x\right)$, we get:

$f^{-1}\left(x\right)=\sqrt[3]{x-6}$

Step 4. verify that the functions are inverses.

Substituting $f\left(x\right)$in localid="1644864594918" $f\left(f^{-1}\left(x\right)\right)=x$, we get:

$$\begin{gathered} f^{-1}\left(f\right(x\left)\right)=x \\ {f}^{-1}(x^3+6))=x \\ \sqrt[3]{x^3+6-6}=x \\ \sqrt[3]{x^3}=x \\ x=x \end{gathered}$$

This is True.

Substituting $f^{-1}\left(x\right)$in $f^{-1}\left(f\right(x\left)\right)=x$, we get:

$$\begin{gathered} f^{-1}(x^3+6)=x \\ \sqrt[3]{x^3+6-6}=x \\ \sqrt[3]{x^3}=x \\ x=x \end{gathered}$$

This is true.

Thus, the obtained inverse is correct